Band 2009, Heft 631

For each positive integer k , we investigate the L -function attached to the k -th symmetric power of the F -crystal associated to the family of cubic exponential sums of x 3 + λx where λ runs over . We explore its rationality, field of definition, degree, trivial factors, functional equation, and Newton polygon. The paper is essentially self-contained, due to the remarkable and attractive nature of Dwork's p -adic theory. A novel feature of this paper is an extension of Dwork's effective decomposition theory when k < p . This allows for explicit computations in the associated p -adic cohomology. In particular, the action of Frobenius on the (primitive) cohomology spaces may be explicitly studied.

The fixed point set under a natural torus action on projectivized moduli spaces of simple representations of quivers is described. As an application, the Euler characteristic of these moduli is computed.

In this paper we construct, using Stark elements of Rubin [Ann. Inst. Fourier 46: 33–62, 1996], Kolyvagin systems for certain modified Selmer structures (that are adjusted to have core rank one in the sense of [Mazur and Rubin, Mem. Amer. Math. Soc. 168: viii, 96, 2004]) and prove a Gras-type conjecture, relating these Kolyvagin systems to appropriate ideal class groups, refining the results of [Rubin, J. reine angew. Math. 425: 141–154, 1992] (in a sense we explain below), and of [Perrin-Riou, Ann. Inst. Fourier 48: 1231–1307, 1998], [Rubin, Euler systems, Princeton University Press, 2000] applied to our setting.

Given our set-up of a system of abstract curves and maps between them satisfying certain assumptions, we prove a classicality criterion for overconvergent sections of line bundles over these curves. As a result we prove such criteria for overconvergent modular forms over various Shimura curves. In particular, we provide a classicality criterion for overconvergent modular forms studied in [Kassaei, Compos. Math. 140: 359–395, 2004] and their higher-level generalizations.

In this paper, we compute all Gram determinants associated to all cell modules of Birman-Wenzl algebras. As a by-product, we give a necessary and sufficient condition for Birman-Wenzl algebras being semisimple over an arbitrary field.

We formulate a strange duality map for parabolic symplectic bundles. We shall prove that as a smooth curve degenerates to a nodal curve, the strange duality map becomes the direct sum of the strange duality maps for parabolic bundles on the normalization of the nodal curve.

We classify codimension two complex submanifolds of projective space X n ⊂ having the property that any line through a general point x ∈ X having contact to order two with X at x automatically has contact to order three. We give applications to the study of the Debarre-de Jong conjecture and of varieties whose Fano variety of lines has dimension 2 n – 4.